# Primitive rational points on expanding horocycles in products of the modular surface with the torus

## TL;DR

This paper establishes effective equidistribution results for primitive rational points and monomial-defined points along long horocycle orbits in products of the torus and the modular surface, answering a previously open question.

## Contribution

It provides the first effective equidistribution results for primitive rational points on expanding horocycles in these product spaces, including joint distribution under congruence conditions.

## Key findings

- Effective equidistribution of primitive rational points on horocycles.
- Joint equidistribution of conjugate rational points in the torus and modular surface.
- Answering an open question from prior joint work.

## Abstract

We prove effective equidistribution of primitive rational points and of primitive rational points defined by monomials along long horocycle orbits in products of the torus and the modular surface. This answers a question posed in joint work by the first and the last named author with Shahar Mozes and Uri Shapira. Under certain congruence conditions we prove the joint equidistribution of conjugate rational points in the two-torus and the modular surface.